Phoenix will host the final game of the first-round series on Tuesday night.
Lauri Korpikoski scored a short-handed goal 4:10 in to spark the win and keep the Coyotes alive.
Mathieu Schneider snapped the Coyotes' 0-for-19 skid on the power play with a goal early in the second for a 2-0 lead, and Radim Vrbata scored with the man advantage midway through the period to restore the two-goal edge.
Wojtek Wolski's even-strength goal put Phoenix ahead 4-1 late in the second.
Jimmy Howard made 24 saves for the Red Wings.
Howard, a finalist for the Calder Trophy — given to the NHL's top rookie — gave up at least four goals for the third time in the series after allowing only one the previous two games as Detroit took a 3-2 lead.
Phoenix went ahead 1-0 on its first shot Sunday. The Coyotes netted their fourth goal late in the second period on their 15th shot and a fifth on their 23rd.
Bryzgalov made 14 saves in the first period and 11 in the second. He was solid in the third as Phoenix cruised to victory.
Detroit's Brad Stuart scored 2:51 into the second — 24 seconds after Schneider put the Coyotes up 2-0 — to fire up the crowd, but the fans were quieted by Vrbata's and Wolski's goals.
Phoenix ended any hopes the Red Wings had for a comeback when Taylor Pyatt scored a power-play goal 5:25 into the third.
Detroit's Darren Helm scored with 3:31 left, but the goal came much too late. The Red Wings missed opportunities, going 0-for-5 on the power play, and couldn't slow down the Coyotes — 3-for-6 with a man advantage.
The Coyotes have been a feel-good story this season after enduring tumultuous times that threatened the team's long-term future in Arizona. Now the Western Conference's fourth-seeded team has a chance to eliminate the defending conference champions with one home win.
Phoenix's previous owner took the franchise into bankruptcy last year, leading to the NHL buying the team in November. Wayne Gretzky resigned as coach nine days before the opener, putting Dave Tippett behind the bench.
The Coyotes, in the postseason for the first time since 2002, are 0-5 in the first round of the playoffs. The franchise hasn't advanced since 1987, when it was the Winnipeg Jets.
The Red Wings are in the playoffs for the 19th straight time and are shooting for another deep run. Detroit has won nine series — including the 2008 Stanley Cup finals — since a first-round exit four years ago in coach Mike Babcock's first season.
Detroit didn't start the postseason with home-ice advantage for the first time since 1991. It hasn't won a Game 7 on the road since a 1964 victory at Chicago in the first round.
The Red Wings blew great chances to get off to a good start Sunday as Phoenix was called for three penalties in the first 4:42. Detroit had a 5-on-3 power play for 1:09, but couldn't capitalize.
The Coyotes were the only team that took advantage as Stuart nonchalantly carried the puck out of his end. Korpikoski snatched it and scored to give Phoenix a 1-0 lead.
Notes
The Coyotes scratched captain Shane Doan, keeping him out of the lineup for the third straight game because of a shoulder injury sustained at Detroit in Game 3. … Former Red Wings star Brendan Shanahan attended the game as a league observer and drew applause when shown on the video boards. Shanahan said he's been in "watch-and-learn mode" since being hired by the NHL four months ago.